Crossing and non-crossing families

TL;DR

This paper investigates the size of crossing and non-crossing families in planar point sets, establishing bounds and algorithms for their detection, thereby advancing understanding of geometric configurations in combinatorial geometry.

Contribution

It introduces a new bound relating crossing and non-crossing families, and provides constructive algorithms with expected polynomial time for finding such families.

Findings

Either a large crossing family or a non-crossing family exists in any point set.

Constructive algorithms can find these families efficiently in expected linear or near-linear time.

The results extend recent theoretical bounds in geometric combinatorics.

Abstract

For a finite set $P$ of points in the plane in general position, a \emph{crossing family} of size $k$ in $P$ is a collection of $k$ line segments with endpoints in $P$ that are pairwise crossing. It is a long-standing open problem to determine the largest size of a crossing family in any set of $n$ points in the plane in general position. It is widely believed that this size should be linear in $n$. Motivated by results from the theory of partitioning complete geometric graphs, we study a variant of this problem for point sets $P$ that do not contain a \emph{non-crossing family} of size $m$, which is a collection of 4 disjoint subsets $P_1$, $P_2$, $P_3$, and $P_4$ of $P$, each containing $m$ points of $P$, such that for every choice of 4 points $p_i \in P_i$, the set $\{p_1,p_2,p_3,p_4\}$ is such that $p_4$ is in the interior of the triangle formed by $p_1,p_2,p_3$. We prove that, for every $m \in \mathbb{N}$, each set $P$ of $n$ points in the plane in general position contains either a crossing family of size $n/2^{O(\sqrt{\log{m}})}$ or a non-crossing family of size $m$, by this strengthening a recent breakthrough result by Pach, Rubin, and Tardos (2021). Our proof is constructive and we show that these families can be obtained in expected time $O(nm^{1+o(1)})$. We also prove that a crossing family of size $\Omega(n/m)$ or a non-crossing family of size $m$ in $P$ can be found in expected time $O(n)$.